You may blanche at the thought of adding colour to foods other than cupcakes, but there are many foods themselves that provide colour.
The main one added here is actually tomato puree, which gives an organgey red colour of course. However, the carotenoid, lycopene, that gives tomato its red colour is oil soluble. That means it will stay in the mix when heated unless it is carried out in oil globules, but certainly any water coming out will not be coloured.
So, if you want a ‘bleeding burger’, where the colour on the outside of the burger is different to the inside (rare cook) and any juices that ooze out are coloured, like blood, then you need a water soluble colour.
This can easily be obtained by adding beetroot powder; it’s a bit pink, so a little yellow balances it towards blood, but do not use turmeric as this is not water soluble.
If you want industrial quantities of natural colours, try: